The paper presents a modular superposition calculus for the
combination of first-order theories involving both total
and partial functions. Modularity means that inferences are
pure, only involving clauses over the alphabet of either
one, but not both, of the theories. The calculus is shown
to be complete provided that functions that are not in the
intersection of the component signatures are declared as
partial. This result also means that if the
unsatisfiability of a goal modulo the combined theory does
not depend on the totality of the functions in the
extensions, the inconsistency will be effectively found.
Moreover, we consider a constraint superposition calculus
for the case of hierarchical theories and show that it has
a related modularity property. Finally we identify cases
where the partial models can always be made total so that
modular superposition is also complete with respect to the
standard (total function) semantics of the theories.